A Note on the Hyperbolicity Cone of the Specialized Vámos Polynomial

Kummer, Mario

doi:10.1007/s10440-015-0036-z

Cited by 11 publications

(12 citation statements)

References 15 publications

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“…The specialized Vámos polynomial is the restriction p G 4 (x 1 , x 2 , x 3 , x 4 ) = p V 8 (x 1 , x 1 , x 2 , x 2 , x 3 , x 3 , x 4 , x 4 ) of p V 8 to a fourdimensional subspace. It is known that no power of p V 8 has a definite determinantal representation [Brä11] and that the same holds for p G 4 [Kum16].…”

Section: All Of the Equality Signs Inmentioning

confidence: 95%

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Saunderson¹

2019

SIAM J. Appl. Algebra Geometry

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We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems-a class of convex optimization problems that generalize semidefinite programs. We show how to produce families of nonnegative polynomials (which we call hyperbolic certificates of nonnegativity) from any hyperbolic polynomial. We investigate the pairs (n, d) for which there is a hyperbolic polynomial of degree d in n variables such that an associated hyperbolic certificate of nonnegativity is not a sum of squares. If d ≥ 4 we show that this occurs whenever n ≥ 4. In the degree three case, we find an explicit hyperbolic cubic in 43 variables that gives hyperbolic certificates that are not sums of squares. As a corollary, we obtain the first known hyperbolic cubic no power of which has a definite determinantal representation. Our approach also allows us to show that, given a cubic p, and a direction e, the decision problem "Is p hyperbolic with respect to e?" is co-NP hard.

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Section: All Of the Equality Signs Inmentioning

confidence: 95%

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Saunderson¹

2019

SIAM J. Appl. Algebra Geometry

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“…This is explained in detail in [14,Proposition 3.8]. Setting α = r/2, the Riesz kernel for det(x) −α is equal to (10) q(y) = π…”

Section: Hyperbolic Polynomialsmentioning

confidence: 99%

“…For general symmetric matrices A i , the fibers L −1 (y) are spectrahedra and not polytopes. If α > m−1 2 then the Riesz kernel exists, and its values are found by integrating (10) over the spectrahedra L −1 (y). In particular, if α = m+1 2 , then the value of the Riesz kernel q(y) equals, up to a constant, the volume of the spectrahedron L −1 (y).…”

Section: Hyperbolic Polynomialsmentioning

confidence: 99%

Positivity Certificates via Integral Representations

Kozhasov¹,

Michałek²,

Sturmfels³

2019

Preprint

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Complete monotonicity is a strong positivity property for real-valued functions on convex cones. It is certified by the kernel of the inverse Laplace transform. We study this for negative powers of hyperbolic polynomials. Here the certificate is the Riesz kernel in Gårding's integral representation. The Riesz kernel is a hypergeometric function in the coefficients of the given polynomial. For monomials in linear forms, it is a Gel'fand-Aomoto hypergeometric function, related to volumes of polytopes. We establish complete monotonicity for sufficiently negative powers of elementary symmetric functions. We also show that small negative powers of these polynomials are not completely monotone, proving one direction of a conjecture by Scott and Sokal.

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“…The specialization studied by Kummer [25] is the restriction of f to the subspace L = {θ ∈ R 8 : θ 1 = θ 2 , θ 3 = θ 4 , θ 5 = θ 6 , θ 7 = θ 8 }. The main result in [25] states that the 3-dimensional body C L is a spectrahedron. The bijection from C L to the interior of K L = C ∨ L factors through the positive variety L ∇f 0 ⊂ RP 7 .…”

Section: Restricting To Linear Subspacesmentioning

confidence: 99%

Exponential varieties

Michałek

Sturmfels

Uhler

et al. 2016

Proc. London Math. Soc.

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Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses of symmetric matrices satisfying linear constraints. This class includes Gaussian graphical models. We develop a general theory of exponential varieties. These are derived from hyperbolic polynomials and their integral representations. We compare the multidegrees and ML degrees of the gradient map for hyperbolic polynomials.

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A Note on the Hyperbolicity Cone of the Specialized Vámos Polynomial

Abstract: The specialized Vámos polynomial is a hyperbolic polynomial of degree four in four variables with the property that none of its powers admits a definite determinantal representation. We will use a heuristic method to prove that its hyperbolicity cone is a spectrahedron.

Cited by 11 publications

References 15 publications

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Positivity Certificates via Integral Representations

Exponential varieties

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